Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,7,8)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258112, #72281479, #72302203, … |
| maximal normal subgroup of | #72258112, #72281479, #72302203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92737);